Itô stochastic differentials

John Armstrong and Andrei Ionescu

Stochastic Processes and their Applications, 2024, vol. 171, issue C

Abstract: We give an infinitesimal meaning to the symbol dXt for a continuous semimartingale X at an instant in time t. We define a vector space structure on the space of differentials at time t and deduce key properties consistent with the classical Itô integration theory. In particular, we link our notion of a differential with Itô integration via a stochastic version of the Fundamental Theorem of Calculus. Our differentials obey a version of the chain rule, which is a local version of Itô’s lemma. We apply our results to financial mathematics to give a theory of portfolios at an instant in time.

Keywords: Itô integration; Differentials in stochastic analysis; Quadratic variation; Continuous semimartingales (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1016/j.spa.2024.104317

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